Fuzzy Logic in Control. René Jager

Transcription

1 Fuzzy Logic in Control René Jager

2 Fuzzy Logic in Control PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus Prof. ir. K.F. Wakker, in het openbaar te verdedigen ten overstaan van een commissie, door het College van Dekanen aangewezen, op maandag 26 juni 995 te :3 uur door RenéJAGER elektrotechnisch ingenieur, geboren te Amsterdam

3 Dit proefschrift is goedgekeurd door de promotor: Prof. ir. H.B. Verbruggen Samenstelling promotiecommissie: Rector Magnificus, Technische Universiteit Delft, voorzitter Prof. ir. H.B. Verbruggen, Technische Universiteit Delft, promotor Prof. dr. ir. E. Backer, Technische Universiteit Delft Prof. dr. ir. J.J. Kok, Technische Universiteit Eindhoven Prof. dr. H. Koppelaar, Technische Universiteit Delft Prof. ir. H.R. van Nauta Lemke, Technische Universiteit Delft Prof. dr. D. Dubois, Université Paul Sabatier Ir. P.M. Bruijn, Technische Universiteit Delft Ir. P.M. Bruijn heeft als begeleider in belangrijke mate aan het totstandkomen van het proefschrift bijgedragen. CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Jager, René Fuzzy logic in control / René Jager. - [S.l. : s.n.]. - Ill. Thesis Technische Universiteit Delft. - With index, ref. - With summary in Dutch. ISBN NUGI 832 Subject headings: fuzzy logic / fuzzy control. Copyright c 995 by René Jager. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the author.

4 Contents Introduction. Why yet another work on fuzzy control? ::::::::::::::::::.2 Why fuzzy control and where does it fit in? :::::::::::::::: 3.3 Fuzzy control and control systems theory ::::::::::::::::: 6.3. Controllers as static functions ::::::::::::::::::: Stability issues ::::::::::::::::::::::::::: 8.4 Relation to artificial and computational intelligence : : : : : : : : : : : : 9.5 What to expect: a road-map for this thesis ::::::::::::::::: 2 Fuzzy sets and relations 3 2. Fuzzy sets ::::::::::::::::::::::::::::::::: What are fuzzy sets? :::::::::::::::::::::::: Properties of fuzzy sets :::::::::::::::::::::: Fuzzy numbers and intervals :::::::::::::::::::: The extension principle :::::::::::::::::::::: Fuzzy set representations ::::::::::::::::::::: Hedges: linguistic modifiers :::::::::::::::::::::::: Powered hedges :::::::::::::::::::::::::: Shifted hedges ::::::::::::::::::::::::::: Scaled hedges ::::::::::::::::::::::::::: Operations on fuzzy sets :::::::::::::::::::::::::: Union and intersection ::::::::::::::::::::::: Complement of fuzzy sets ::::::::::::::::::::: 33 i

5 ii Contents 2.4 Fuzzy relations ::::::::::::::::::::::::::::::: Projection and cylindrical extension :::::::::::::::: Composition of fuzzy relations :::::::::::::::::: Summary and remarks ::::::::::::::::::::::::::: 42 3 Fuzzy logic and reasoning Fuzzy propositions ::::::::::::::::::::::::::::: Logical connectives :::::::::::::::::::::::: Negation in fuzzy propositions :::::::::::::::::: Fuzzy rules ::::::::::::::::::::::::::::::::: Representation of a fuzzy rule ::::::::::::::::::: Fuzzy implications ::::::::::::::::::::::::: Aggregation of fuzzy rules ::::::::::::::::::::: Classification of fuzzy implications :::::::::::::::: Rule base properties :::::::::::::::::::::::: Continuity of a rule base :::::::::::::::: Consistency of a rule base :::::::::::::::: Completeness of a rule base ::::::::::::::: Fuzzy reasoning :::::::::::::::::::::::::::::: Inference of a fuzzy rule :::::::::::::::::::::: Compositional rule of inference ::::::::::::: Generalized modus ponens and tollens ::::::::: Criteria for generalized modus ponens ::::::::: Inference of a rule modeled by a T-implication ::::: Inference of a fuzzy rule base ::::::::::::::::::: Local versus global inference : : : : : : : : : : : : : : Rules modeled by classical-conjunction-based implications ::::::::::::::::::::::::::: Rules modeled by classical-implication-based implications ::::::::::::::::::::::::::: Summary and remarks ::::::::::::::::::::::::::: 74 4 Fuzzy control Theoretical approach to fuzzy control ::::::::::::::::::: Fuzzification of inputs ::::::::::::::::::::::: Defuzzification of output ::::::::::::::::::::: Center-of-gravity defuzzification : : : : : : : : : : : : Indexed defuzzification methods : : : : : : : : : : : : Mean-of-maxima defuzzification : : : : : : : : : : : : Center-of-area defuzzification method ::::::::: Example of theoretical approach :::::::::::::::::: 83

6 Contents iii 4.2 Practical approach to fuzzy control :::::::::::::::::::: Fuzzy inference in practice :::::::::::::::::::: Practical fuzzy inference scheme : : : : : : : : : : : : Inference with T-implications : : : : : : : : : : : : : : Inference with S-implications : : : : : : : : : : : : : : Inference with other implications : : : : : : : : : : : : Input fuzzification ::::::::::::::::::::::::: Common inference methods ::::::::::::::::::: Max-min method :::::::::::::::::::: Max-prod method :::::::::::::::::::: Sum-prod method :::::::::::::::::::: Defuzzification in practice ::::::::::::::::::::: Averaging defuzzification methods ::::::::::: Height-related methods ::::::::::::::::: Extended defuzzification methods :::::::::::7 4.3 Fuzzy control rules ::::::::::::::::::::::::::::: Mamdani fuzzy rules ::::::::::::::::::::::: Sugeno fuzzy rules ::::::::::::::::::::::::: Differences and similarities ::::::::::::::::::::7 4.4 Fuzzy linear control :::::::::::::::::::::::::::: Fuzzy linear models :::::::::::::::::::::::: Fuzzy linear controllers :::::::::::::::::::::: Experiments with fuzzy pole-placement controller :::::::: Remarks and considerations :::::::::::::::::::: Fuzzy controller as input-output mapping ::::::::::::::::: Fuzzy system as universal approximator : : : : : : : : : : : : : : Linear controller fuzzy controller :::::::::::::::: Fuzzy controller analysis :::::::::::::::::::::::::: Role of fuzzy sets ::::::::::::::::::::::::: Number of fuzzy sets :::::::::::::::::: Overlapping fuzzy sets ::::::::::::::::: Shape of fuzzy sets ::::::::::::::::::: Fuzzy sets for the output :::::::::::::::: Role of operators ::::::::::::::::::::::::: Negation in rule premises :::::::::::::::: Logical and connective ::::::::::::::::: Logical or connective :::::::::::::::::: Role of the rule base :::::::::::::::::::::::: Incompleteness and interpolation : : : : : : : : : : : : Exceptions and rule precedence ::::::::::::: Conclusions and remarks :::::::::::::::::::::::::44

7 iv Contents 5 Adaptive fuzzy control Self-organizing fuzzy control ::::::::::::::::::::::: Self-organizing controller scheme ::::::::::::::::: Relation-based approach :::::::::::::::::::::: Numerical example of the relation-based approach : : : Rule-based approach :::::::::::::::::::::::: Simplified rule-based approach :::::::::::::::::: Fuzzy relations as associative memories :::::::::::::::::: Adaptation by fuzzy supervisors :::::::::::::::::::::: Fuzzy supervised PID-control ::::::::::::::::::: Adaptive fuzzy expert controller :::::::::::::::::: Gradient-descent adaptation :::::::::::::::::::::::: The basic adaptation scheme :::::::::::::::::::: Restrictions on adaptation ::::::::::::::::::::: Maintaining fuzzy partitions :::::::::::::::::::: Comparison with other learning systems :::::::::::::::: Relation to radial-basis function networks ::::::::::::: Comparison with generalized CMAC :::::::::::::::8 5.6 Conclusions and remarks :::::::::::::::::::::::::82 6 Fuzzy logic in knowledge-based systems Knowledge-based systems for control ::::::::::::::::::: Knowledge representation ::::::::::::::::::::: Real-time control requirements :::::::::::::::::: Possibility theory ::::::::::::::::::::::::::::: Possibility distributions :::::::::::::::::::::: The concept of a possibility distribution :::::::: Fuzzy sets and possibility distributions ::::::::: Different interpretations of propositions :::::::: Possibility and necessity measures ::::::::::::::::: Principles of minimum and maximum specificity ::::::::: Principle of minimum specificity : : : : : : : : : : : : Principle of maximum specificity : : : : : : : : : : : : Rules and conditional possibility distribution ::::::::::: Approximate reasoning :::::::::::::::::::::::::: Reasoning modes ::::::::::::::::::::::::: Translation rules :::::::::::::::::::::::::: Quantification rules ::::::::::::::::::: Qualification rules :::::::::::::::::::: Practical considerations :::::::::::::::::::::: Reasoning with possibility distributions ::::::::::::::::::26

8 Contents v 6.4. Interpretation of rules ::::::::::::::::::::::: Possibility-qualifying rules ::::::::::::::: Certainty-qualifying rules :::::::::::::::: Truth-qualifying rules :::::::::::::::::: Fuzzy control rules in terms of rule types :::::::: An inference break-up method ::::::::::::::::::: Breaking up the inference :::::::::::::::: Reduction of inference break-up : : : : : : : : : : : : Summary of inference break-up ::::::::::::: Other and derived approaches to fuzzy reasoning ::::::::::::: Reasoning with fuzzy truth values ::::::::::::::::: Baldwin s method :::::::::::::::::::: Tsukamoto s method :::::::::::::::::: Mizumoto s method ::::::::::::::::::: Fuzzy reasoning based on similarity measures ::::::::::: Yager s method ::::::::::::::::::::: Turksen and Zhong s method : : : : : : : : : : : : : : Reasoning with domain scaling ::::::::::::: Reasoning with linguistic qualifiers :::::::::::::::: Remarks and considerations :::::::::::::::::::: Conclusions and remarks :::::::::::::::::::::::::234 7 Conclusions and suggestions 237 A Fuzzy logic operators 243 B Linear controller fuzzy controller proof 247 C Derivation of restricted learning rule 25 D GCMAC: Generalized Cerebellar Model Articulation Controller 255 E RICE: Routines for Implementing C Expert systems 259 E. The inference engine and supporting tools :::::::::::::::::259 E.2 Examples using RICE in simulation and control : : : : : : : : : : : : : :262 F Proofs for inference break-up method 267 F. Rule break-up :::::::::::::::::::::::::::::::267 F.2 Rule inference break-up ::::::::::::::::::::::::::268 F.3 Rule base inference break-up :::::::::::::::::::::::269 List of symbols 285

9 vi Contents List of abbreviations 289 Summary 29 Vage logica in de regeltechniek 295 Curriculum vitae 299 Acknowledgments 3 Author index 33 Subject index 37

10 Introduction Nowadays, fuzzy control is a hot topic. Journals and books on fuzzy set theory are published, symposia on fuzzy modeling and control are organized and software packages for fuzzy control are released. Despite all the publications on fuzzy sets and fuzzy control this thesis has been written, because, in our opinion, there exists much misunderstanding about fuzzy control. This thesis aims at providing an analysis of fuzzy control to clear this up this misunderstanding. The first section of this chapter describes the reasons for writing this thesis in more detail. The second section briefly addresses the place of fuzzy control within the field of control. In this thesis a controller is considered as a (static) function. This functional approach to control is discussed in section.3.. Stability issues are briefly addressed in section.3.2. The place of fuzzy logic and fuzzy control within the field of artificial, or computational, intelligence is addressed in section.4. The last section can be used as a road-map for this thesis: it helps you to avoid areas in the field of fuzzy control that you are already familiar with, and it can serve as a guide to direct you to areas of your own interest.. Why yet another work on fuzzy control? When reaching for this thesis, one might ask: Why yet another thesis on fuzzy control?. Considering the available amount of books, journals and conference proceedings, this question might seem legitimate. However, on one hand, many books and journals are very mathematical and are not focused on fuzzy control. On the other hand, many books and

11 2 Introduction articles only focus on simple fuzzy control problems and applications, and are incomplete with respect to the theoretical frameworkin which fuzzy control resides. The contributions that do focus on fuzzy control, including the contributions to the enormous number of symposia and workshops on fuzzy systems, do, in our opinion, not place fuzzy control in the right perspective. There is still a lot of vagueness and misunderstanding around the topic, which, we think, is not necessary. Thus, this more or less answers the question why this thesis is written: a demystification of fuzzy control and on the same time a profilation of fuzzy control. The aim of this is to place fuzzy control in the right perspective. For this reason, only half of this thesis focuses on fuzzy control and the other half is used for underlying concepts, necessary for a good understanding of fuzzy control, and more general topics related to fuzzy logic and fuzzy control. Nowadays books are being published which are written for people not yet or barely familiar with fuzzy logic and fuzzy control. Fuzzy control is treated as such, without questioning the underlying concepts. However, those underlying concepts are very important, as will be shown in this thesis, and can lead to conclusions that suggest the elimination of fuzzy calculus within fuzzy control and use well-known interpolation techniques: in that case the fuzzy part is limited to user-interfacing. In the field of fuzzy control many software tools are available and are mostly promoted with slogans which state that fuzzy control is the solution to all our problems. Considering this hype around fuzzy control and the resulting misconception is reason enough to take a close look at what a fuzzy controller actually is and how it works. Approximate reasoning, based on fuzzy set theory and possibility theory, provides several techniques to reason with fuzzy and uncertain concepts in knowledge-based systems. Applying fuzzy techniques in knowledge-based systems can provide a knowledge representation and inference which is closer to the way humans express their knowledge and reason with it than in the case of conventional knowledge-based systems based on classical logic. This comes very close to the field of natural language understanding and processing. It is hard to imagine oneself talking with someone else which cannot distinguish between fairly true and very true, although both qualifications are rather vague. Fuzzy logic and approximate reasoning enable us to (partly) model human reasoning by means of computer implementations. When we take a look into the future and imagine humans communicating with computers on a, compared to nowadays, high level of intelligence, then this requires the modeling of human reasoning and natural language. Approximate reasoning provides a theoretical framework to perform this modeling. Fuzzy control can be regarded as a small part within the framework of approximate reasoning. For this reason a chapter is dedicated to fuzzy logic in knowledge-based systems, extending the narrow view used in fuzzy control literature to the broader framework where fuzzy control resides in.

12 .2 Why fuzzy control and where does it fit in? 3.2 Why fuzzy control and where does it fit in? Today, a lot of interest from industry in fuzzy systems can be noticed. In western countries, this is currently mostly limited to an orientation to the field of fuzzy control. One of the reasons for this orientation towards fuzzy control is because competing companies (mostly Japanese) are using or starting to use fuzzy control in competing products and advantages of doing so are reported in literature. Many examples of fuzzy control applications exist in consumer product. Hence, the growing interest in fuzzy control is understandable, but the question then rises why do the (competing) companies use fuzzy control? Considering the discussions in literature and on newsgroups and mailing lists y the following reasons can be extracted:. Fuzzy control is a new technology and therefore can be used to avoid patentclaims of similar solutions for technical problems, which are based on a different technique. 2. Nowadays, in Japan, fuzzy is wanted by consumers, since it represents hightech. In this case fuzzy techniques are mostly used as a marketing tool. 3. The development of fuzzy controllers is easier to learn and requires less skilled personnel than the development of conventional controllers. This results in cheaper production. 4. Fuzzy controllers provide more robustness than conventional control. 5. Fuzzy controllers are more appropriate to control nonlinear processes. For academia, reasons and 2 should not play a role, so in the following we will focus on the last three reasons. Fuzzy controllers are represented by if-then rules and thus can provide a user-friendly and understandable knowledge representation. One can see this as a (very) high-level programming language, where the program consists of if-then rules and the compiler and/or interpreter results in a nonlinear control algorithm. Hence, programming by means of qualitative statements, represented by means of if-then statements, to obtain a program working on quantitative domains, provided by sensor and actuator signals. Intuitively, this entails loss of information, because there is no unique translation from a qualitative entity to a quantitative representation except for some special cases. For example, there The main newsgroup on fuzzy systems is comp.ai.fuzzy. y A major mailing list in this field is fuzzy-mail@vexpert.dbai.tuwien.at,which also mirrors newsgroup comp.ai.fuzzy.

13 4 Introduction is no unique translation from large voltage to a real-valued voltage and vice versa. Because in control the results of a controller are expected to be precise quantities since those results are signals for actuators of motors, valves, pumps, heaters, etc., special additional techniques are necessary for the translation of qualitative information to quantitative information. It is an advantage though, that (complex) control strategies which are known by operators or process engineers in the form of experience and/or domain knowledge, can be programmed and maintained in a user-friendly and understandable way. It is often claimed that fuzzy control provides more robustness. However, no research results has been found that prove that fuzzy controllers are more robust than conventional controllers in general. As will be shown in this thesis, a fuzzy controller is in fact a static nonlinearity and whether this is more robust than a conventional controller depends on the rules defining this static nonlinearity. However, when the variations of process parameters are (partly) known, a fuzzy controller can be designed to be less sensitive for those parameter changes and thus be more robust than a comparable linear controller. Indeed, this can be compared to gain-scheduling (Åström and Wittenmark, 984) with the difference that a fuzzy controller implicitly provides bumpless transfers from one set of controller parameters to another. So, fuzzy controllers are more robust should be interpreted as fuzzy controllers can be more robust to known parameter changes. How to device a fuzzy controller which is more robust still remains a problem since it depends mainly on knowledge of the process to be controlled. Another claim often made about fuzzy control is that fuzzy control is more appropriate to control nonlinear processes. Whether a fuzzy controller, or a nonlinear controller in general, is in principle able to control a nonlinear process sufficiently, depends primarily on the chosen inputs of the controller. Fuzzy controllers are often said to be superior to their corresponding linear controller to control nonlinear processes. For fuzzy PID-like controllers, this is only true for a small set of problems, namely when the nonlinearity of the process can be written as a function of the error and its derivatives, being the input signals of the controller. This is normally not the case, because the error and error change are not only determined by the process, but also by the externally defined reference signal. If it is desired that a controller is able to capture nonlinearities of the process to be controlled, the controller should not be based on the error and its derivatives: one could use, for example, the reference signal or the process output (and their derivatives) as additional inputs of the controller. In general it can be stated, based on the same controller inputs, that: a fuzzy controller can control a nonlinear process as least as good as its corresponding linear controller can do, just because a fuzzy controller can control a linear process as least as good as its corresponding linear controller can do. Thus, in principle, a fuzzy controller is more

14 .2 Why fuzzy control and where does it fit in? 5 capable of controlling a nonlinear process, but additional knowledge of the nonlinearities of the process is needed. Summarizing the previous discussion, we can state the following: fuzzy control provides a method to construct controller algorithms in a user-friendly way and provides the ability to capture the nonlinear control behavior of humans which has proven to be appropriate for many complex tasks. Having a design method for controllers which is closer to human thinking and perception can reduce development time and requires less skilled personnel to design controllers. The economical benefit of this is trivial. It should be noted that the robustness of human controllers is primarily due to their ability to adapt to a changing environment and their learning capability. Building this ability into fuzzy controllers is beyond today s application of fuzzy control in consumers products, but research on adaptive fuzzy control has been done for quite some time. When the current state of fuzzy control is considered, it can be stated that the main areas in which fuzzy control can be applied, are the following:. Processes which can be adequately controlled by humans and the controller to be designed has sensors to provide similar information used by humans to control the process. Examples are the application of fuzzy logic in automatic transmission for cars, washing machines, etc. Nowadays, there are many applications of fuzzy logic in consumer products. 2. Processes which are currently controlled by (basically) linear control algorithms and need further development resulting in nonlinear control algorithms which are known by operators or process engineers. Mamdani (994) states that: Fuzzy logic is successful because it replaces the classical PID controller. When tuned, the parameters of a PID controller affect the shape of the entire control surface. Because fuzzy logic control is a rule-based controller, the shape of the control surface can be individually manipulated for the different regions of the state space, thus limiting possible effects to neighboring regions only. As a starting point for a fuzzy controller, the linear controller that is currently used to control the process in question can be used, because, under certain conditions (section 4.5.2), a fuzzy controller can be designed to emulate a linear controller. A more or less critical point of view on fuzzy control is given by Elkan (994), who states that fuzzy controllers are characterized by the following properties: fuzzy controllers use typically fewer than rules; often even fewer than 2 rules;

15 6 Introduction the knowledge within a fuzzy controller is usually shallow, both statically and dynamically; the knowledge within a fuzzy controller typically reflects correlations between controller inputs and outputs; the numerical parameters of a fuzzy controller are tuned in a learning process; fuzzy controllers use fuzzy logic operators. Elkan (994) states that the success of fuzzy control is mainly because of the first four properties and that the use of fuzzy logic is not essential. In our opinion these statements about fuzzy controllers are mostly correct, but it should be expressed that these properties do not have to be interpreted as being negative, since it allows user-friendly development of (nonlinear) controllers. Although the rest of Elkan s provocative article does not show a deep understanding of fuzzy logic and related topics in our opinion, it started a discussion on fuzzy logic and fuzzy control which contributes to further discussion of the strengths and limitations of fuzzy logic (Zadeh, 994b)..3 Fuzzy control and control systems theory In this section we describe the view on control that is used in this thesis. An important issue is the way a controller is considered. This is described in the first subsection. The second subsection addresses briefly the aspect of stability..3. Controllers as static functions Today, most controllers are implemented by computer algorithms. This implies that the controller inputs are measured at certain sampling rates. For example, the linear part of a classical PID controller can be represented by: u(t) =K P e(t)+k I Z t e( )d + K D de(t) dt (.) where u(t) is the control signal fed to the process to be controlled and e(t) is the error signal: the difference between the desired and measured process output. A computer implementation of a PID controller can be expressed as a difference equation: u PID [k] =u PID [k, ] + k I e[k]+k P e[k]+k D 2 e[k] (.2) Proportional-Integral-Differential.

16 .3 Fuzzy control and control systems theory 7 with: e[k] = e[k], e[k, ] 2 e[k] =e[k], e[k, ] When we consider a PI or PD controller, the following difference equations can be derived: u PI [k] = u PI [k, ] + k I e[k]+k P e[k] u PD [k] =k P e[k]+k D e[k] (.3a) (.3b) Equations (.2), (.3a) and (.3b) can be compared with the algebraic representations of a (hyper)plane: y = a + nx i= a i x i (.4) A schematic representation for (.3a) and (.3b) is shown in figure.. When we consider a mapping from controller inputs to controller outputs in general (MIMO system), the controller function is represented by a mapping: y = f(x) (.5) This is the way controllers are considered in this thesis: controller outputs are static functions (mappings) of the controller inputs. Dynamical behavior of a controller, like differential or integral action, are emulated by extending the controller function to more inputs. Those inputs are delays or differences of other inputs and outputs. Hence, a controller is considered to consist of a static controller function and additional prefiltering and postfiltering parts to obtain delayed inputs, input differences, integrations, limited signals, etc. Note that this approach is generally also used in the field of neural networks. To be consistent with the above described functional view of controllers, the variable naming in the rest of this thesis will be conform (.5). Hence, controller output(s) will be addressed as y s and input(s) will be addressed as x s. Moreover, this notation is used for fuzzy systems in general in this thesis. Using the same variable naming convention for both models of processes and controllers is because the functional behavior of a model or a controller is similar in our opinion; they both have to fit a certain (non)linear mapping of inputs to outputs, meeting a number of predefined criteria. Multiple-Inputs-Multiple-Outputs.

17 8 Introduction x z, y = + x 2,z, x 3 3X i= a i x i y process Figure.: Example of PI and PD controllers regarded as a static mapping using prefiltering and postfiltering blocks. Parameters are chosen as follows: a =, a 2 = k I and a 3 = k P for a PI controller, and a =, a 2 = k P and a 3 = k D for a PD controller..3.2 Stability issues Leaving the aspect of stability out of a thesis on control is hardly possible. Therefore the stability issue is addressed in this section, but it is also the only place where it is addressed. This is based on two reasons. Firstly, fuzzy controllers can be regarded as nonlinear controllers and for this reason it is difficult to obtain general results on the analysis and design of fuzzy controllers (Driankov et al., 993). The second reason is clearly expressed by Mamdani (993): Industry has never put forward a view that the mathematical stability analysis is a necessary and sufficient requirement for the acceptance of a well designed control system. That is merely the view that control system scientists wished to put forward, but it has never gained currency outside academic circles. :::Prototype testing is more important than stability analysis; stability analysis by itself can never be considered a sufficient test. Moreover, in any practically useful methodology, a stability analysis step would need to be made a desirable but an optional step; it cannot be a necessary step. These statements might seem rather strong and indeed Mamdani was much criticized for these statements, but in our opinion his statements contain a lot of truth. The stability proofs for fuzzy controllers found in literature are restricted to the cases where fuzzy

18 .4 Relation to artificial and computational intelligence 9 controllers are simple, for example PID-like fuzzy controllers, and where the process to be controlled is stable itself; see for example the article of Malki, Li and Chen (994). In most cases the stability proofs are trivial due to the simplicity of the controller and process. If the process cannot be modeled mathematically, for example the control of cement kilns (Östergaard, 99), then stability proofs cannot be given at all. Because the first applications of fuzzy control were controllers for processes which could not (and cannot) be modeled mathematically, like cement kilns, the criticism that fuzzy control does not allow stability analysis was not valid since stability analysis is based on a mathematical model of the process and such models were not available (Mamdani, 993). Considering the numerous applications of fuzzy control in consumer products, one can question the need for a mathematical stability analysis for these, in this case rather simple, control problems. We conclude with another quote from Mamdani (993), because his paper Twenty years of fuzzy control: experiences gained and lessons learnt contains a view on control which closely resemblances ours, including the following statement: Stability is still an important issue but a different way has to be found to study it. In the final analysis all one may be able to do is to build prototypes for the purpose of approval certification. This is a well tried and tested approach used in industry and there is no reason why it may not suffice with control systems as well..4 Relation to artificial and computational intelligence Fuzzy logic is regarded as one of the artificial intelligence (AI) techniques, from which conventional expert systems, neural networks and genetic algorithms are well known. It can be disputed whether, for example, neural networks and genetic algorithms should be considered as artificial intelligence techniques. Zadeh (994a) proposes the denomination soft computing to address the field of neural networks, genetic algorithms, fuzzy logic and combinations of those. Today, the field of fuzzy control and modeling is in many publications considered to overlap with the field of neural networks. Many publications on these neuro-fuzzy systems or fuzzy neural networks can be found in literature. In our opinion this overlap of neural networks and fuzzy systems is purely based on functional equivalence and not based on the underlying ideas. However, considering this functional equivalence and the many publications addressing the merging of these techniques, it seems valid to address those techniques by one name. An important concept of fuzzy set theory and fuzzy logic is the linguistic variable (Zadeh, 994a). In their survey, Dubois and Prade (99) state that the main motivation of fuzzy set theory is apparently the desire to build up a formal, quantitative framework that

19 Introduction captures the vagueness of human knowledge as it is expressed via natural languages. From this point of view, fuzzy logic can be regarded as part of the field of artificial intelligence, because fuzzy logic and approximate reasoning (see chapter 6) can provide a framework for natural language understanding and processing, and modeling of the way humans reason and communicate. Although many authors claim fuzzy logic to be a suitable framework for dealing with uncertainty in expert systems, one can hardly find a real application of such a system (Elkan, 993). As opposed to the large number of theoretical papers on fuzzy logic in expert systems, there exist only a small number of reported prototypes of such systems (Graham, 99). But what about the large number of reported applications of fuzzy control in consumer products? In our opinion these applications are not successful because they are applications of AI, but because they provide a user-friendly method to implement nonlinear controller functions. In other words: Fuzzy controllers have encountered great success by providing an efficient way of implementing an interpolative mechanism, not only in small, but also in very large and complex problems (Dubois et al., 994). Zadeh (994b) expressed the comparison between (fuzzy) control and more general knowledge-based systems as follows: Basically, what differentiates control applications from knowledge-based systems applications is that in control the main problem that has to be addressed is that of imprecision. By contrast, in the case of knowledge-based systems, one has to come to grip with both imprecision and uncertainty. This becomes also clear when one notices that fuzzy control can be considered as only a small part of the theoretical framework of approximate reasoning..5 What to expect: a road-map for this thesis Before providing a road-map for this thesis, it should be stated that this thesis does not contain parts which describe applications of fuzzy control for a specific industrial or laboratory set-up. This is considered not necessary since there exist numerous commercial applications of fuzzy control in consumer products as already pointed out in section.2. When appropriate, small examples are used to clarify described methods or algorithms. The aim of this thesis is to investigate the fundamentals of fuzzy control and to provide an insight in the underlying theory, not to show the success in some application areas. In the remainder of this section a road-map to this thesis is presented. This makes it possible for the reader to exclude parts through which he or she already is already familiar with, or directly go to parts of personal interest. Before giving a short description of the contents of following chapters in this thesis, it should be noticed that each chapter has a final section in which conclusions and/or a short summary are given. Most chapters also

20 .5 What to expect: a road-map for this thesis contain a section which addresses practical aspects, mainly based on issues concerning computer implementation. The next chapter describes partly the theory of fuzzy sets. This includes the basic notion of a fuzzy set, properties of fuzzy sets and operations on fuzzy sets. The aim of this chapter is not to be complete, but to provide the parts of fuzzy set theory which are necessary to understand the remainder of this thesis. Chapter 3 will address the basics of fuzzy logic and reasoning. This assumes that the reader is familiar with fuzzy set theory (chapter 2). Different logical operations will be described and discussed, including logical connectives and implications. Reasoning with fuzzy logic is described for single rules and sets of fuzzy rules (rule bases). Fuzzy control is described and discussed in chapter 4. A detailed description is given as well as advantages and disadvantages of different approaches. In this chapter fuzzy control is considered to be based on fuzzy rules which directly connect controller inputs to controller outputs ( flat rule base). This type of fuzzy control is mostly applied and chaining of fuzzy rules is not considered. An analysis of fuzzy controllers with respect to different controller parameters is provided. Adaptive fuzzy control is an extension of fuzzy control and is described in chapter 5. Two main approaches within the field of adaptive fuzzy control can be distinguished: approaches based on the self-organizing controller introduced by Procyk and Mamdani (979) and approaches based on gradient-descent adaptation. Adaptive fuzzy controllers or models based on the latter approach are often referred to as fuzzy neural networks or neuro-fuzzy systems. Both types are addressed in this chapter. In chapter 6 approximate reasoning is addressed. Approximate reasoning provides a framework to model natural language understanding and human reasoning. Within this field of research one can distinguish several points of view and approaches, which will be described in this chapter. The practical application of approximate reasoning is not always straightforward and difficulties encountered are also discussed in this chapter. The basics of possibility theory, as part of the approximate reasoning framework, are explained in this chapter, but a good understanding of fuzzy set theory and fuzzy logic (chapters 2 and 3) is assumed. The last chapter gives conclusions based on the contents of this thesis, general remarks on addressed topics and suggestions for further research.

21 2

22 2 Fuzzy sets and relations This chapter contains the basics of fuzzy set theory that are necessary for a correct understanding of the rest of this thesis. If the reader is already familiar with the field of fuzzy set theory, this chapter will probably contain nothing new. If the reader is not familiar with the field it will serve as an introduction to fuzzy set theory. This chapter will start with a section about what fuzzy sets are and how they are related to classical (ordinary) set theory. After this, a number of properties of fuzzy sets are given in section A special type of fuzzy set, referred to as fuzzy numbers, is described in section The extension principle is one of the basic concepts in fuzzy set theory and allows mathematical concepts to be extended for use with fuzzy sets, and is addressed in section The union and intersection of fuzzy sets, and the complement of fuzzy sets, are presented in section 2.3. Linguistic modifiers, usually referred to as hedges, are addressed in section 2.2. Section 2.4 deals with fuzzy relations. A final section summarizes this chapter. 2. Fuzzy sets Zadeh (965) introduced fuzzy sets, although the underlying idea or ideas close to it had already been recognized earlier by others, mainly by philosophers. A comprehensive 3

23 4 Fuzzy sets and relations overview is given in the introduction of the readings in Fuzzy Sets for Intelligent Systems, edited by Dubois, Prade and Yager (993). Fuzzy sets as defined by Zadeh are described in this section. In addition to a basic description and some examples, properties of fuzzy sets are addressed (section 2..2). Section 2..3 describes a special type of fuzzy set: fuzzy numbers and intervals. Extending mathematical operations to operate on fuzzy sets is possible by applying the extension principle, which is described in What are fuzzy sets? Classical set theory is well known, and in the field of fuzzy set theory is usually called classical set theory instead of just set theory. The membership A (x) of x of a classical set A, as subset of the universe X, is defined by: A (x) = ( ; iff x 2 A ; iff x 62 A (2.) This means that an element x is either a member of set A ( A (x) =)or not ( A (x) =). Classical sets are also referred to as crisp sets. For many classifications, however, it is not quite clear whether x belongs to a set A or not. For example, if set A represents PCs which are too expensive for a student s budget, then it is obvious that this set has no clear boundaries. Of course, it could be said that a PC priced at $25 is too expensive, but what about PCs priced at $2495 or $252? Are those PCs too expensive or not? Clearly, a boundary could be determined above which a PC is too expensive for the average student, say $25, and a boundary below which a PC is certainly not too expensive, say $. Between those boundaries, however, there remains a vague interval in which it is not quite clear whether a PC is too expensive or not. In this interval, a grade could be used to classify the price as partly too expensive. This is where fuzzy sets come in: sets of which the membership has grades in the interval [,]. A fuzzy set, introduced by (Zadeh, 965), is a set with graded membership in the real interval: A (x) 2 [; ]. A fuzzy set A, a fuzzy subset of X, is denoted by: A = mx i= A (x i )=x i (2.2a) = A (x )=x + + A (x m )=x m (2.2b) where A (x) is known as the membership function, and where X is known as the universe of discourse. When X is not finite, a fuzzy set A is defined by: A = Z X A (x)=x (2.2c)

24 A(x) 2. Fuzzy sets 5 In this thesis the latter is primarily used to denote fuzzy sets. A simple example of a fuzzy set is one representing PCs too expensive for a student s budget. This fuzzy set is depicted in figure 2.. One can see that if the price is below $ the PC is certainly not too expensive, and if the price is above $25 the PC is fully classified as too expensive. In between, an increasing membership of the fuzzy set too expensive can be seen. It is not necessary that the membership linearly increases with the price, nor that there is a discontinuous transient for $ and $25, as will be seen later in this thesis. The choice of the membership function of the fuzzy set is arbitrary. too expensive $ $25 x Figure 2.: Fuzzy set A representing PCs too expensive for a student s budget. The price is represented by variable x Properties of fuzzy sets In this section a number of properties of fuzzy sets is given. The aim of this section is not to be complete, but to provide those parts of fuzzy set theory necessary to make the rest of this thesis understandable. It is mostly a list of commonly used fuzzy set properties. The height of a fuzzy set A, hgt(a), is defined by: hgt(a) = sup A (x) (2.3) x2x

25 A(x) 6 Fuzzy sets and relations and fuzzy sets with a height equal to are called normal. Fuzzy sets called subnormal are characterized by hgt(a) <. The core of a fuzzy set, also referred to as kernel or nucleus, is a crisp subset of X: core(a) =fx 2 X j A (x) =g (2.4) The support of a fuzzy set is also a crisp subset of X: supp(a) =fx 2 X j A (x) > g (2.5) If the support of a fuzzy set is finite, it is called compact support. Figure 2.2 shows schematically the height, core and support of a fuzzy set. height core x support Figure 2.2: Height, core and support of a fuzzy set. The elements of x where A (x) = 2 is defined by: are called crossover points. The-cut of a fuzzy set -cut(a) =fx 2 X j A (x) g (2.6) An -cut of a fuzzy set is often referred to as a level set. Astrong -cut is defined by: -cut(a) =fx 2 X j A (x) >g (2.7)

26 2. Fuzzy sets 7 Thus the core of a fuzzy set can be defined by an -cut, with =: core(a) =-cut(a) and the support of a fuzzy set can be defined by a strong -cut (-cut), with =: supp(a) =-cut(a) Another property of fuzzy sets used in this thesis is whether a fuzzy set is convex or not. A convex fuzzy set is characterized by: 8 x ;x 2 ;x 3 2 X; x x 2 x 3! A (x 2 ) min( A (x ); A (x 3 )) (2.8) where x, x 2 and x 3 are values in X. Hence, the fuzzy set in figure 2.2 is convex. Convexity of a fuzzy set can play an important role when analysing a fuzzy controller in combination with another important property of fuzzy sets: whether the fuzzy sets form a fuzzy partition. When N A fuzzy sets A j are fuzzy subsets of universe X, such a tuple of fuzzy sets (A ;:::;A j ;:::;A NA ) is called a fuzzy partition when: 8x 2 X; XN A j= Aj (x) = (2.9) provided that A j 6= ; and A j 6= X. An example of a fuzzy partition is given in figure 2.3. A fuzzy partition formed by fuzzy sets which are normal and convex, does not contain more than two overlapping fuzzy sets Fuzzy numbers and intervals A fuzzy number is a special type of fuzzy set. A fuzzy set F is a fuzzy number, usually a fuzzy subset of R, if it meets the following criteria: the fuzzy set is convex, as defined by (2.8); the fuzzy set is normalized: hgt(f )=; the membership function of the fuzzy set is piecewise continuous; the core of the fuzzy set consists of one value only.

27 5(x) [2;7](x) A(x) 8 Fuzzy sets and relations x Figure 2.3: A fuzzy partition x x (a) about 5 (b) from about 2 to about 7 Figure 2.4: Fuzzy sets representing fuzzy number about 5 and fuzzy interval from about 2 to about 7.

28 2. Fuzzy sets 9 Thus a fuzzy number is always a fuzzy set, but a fuzzy set is not always a fuzzy number. An example of a fuzzy number about 5 is shown in figure 2.4a. Mathematical operations like addition, subtraction, etc. can be extended for use with fuzzy numbers by means of the extension principle, which is addressed in section In addition to fuzzy numbers one can consider fuzzy intervals (Dubois and Prade, 988). A fuzzy interval is a fuzzy set with the same restrictions as these defined for fuzzy numbers, but with the exception that the core is no longer restricted to one point only. An example of the fuzzy interval from about 2 to about 7 is shown in figure 2.4b The extension principle The extension principle was introduced by Zadeh (975) and is one of the most important elements of fuzzy set theory. As Dubois and Prade (98) put it: it provides a general method for extending non-fuzzy mathematical concepts in order to deal with fuzzy quantities. The extension principle allows the extension of a mapping f from points in X to fuzzy subsets of X: f(a) =f( =x + + n =x n ) (2.a) 4 = =f(x )+ n =f(x n ) (2.b) A simple example is given by the following. Consider the fuzzy set about 5 with a discrete universe, x i 2 Z and mapping f representing the square. Then the application of the extension principle results in: ( about 5 ) 2 =( 2 =4+=5 + 2 =6)2 =( 2 =6+= =36) The extension principle applied to a function or mathematical operation f(x ;:::;x n ) is defined by: B (y) = sup A (x ;:::;x n ) x ;:::;x n y=f(x ;:::;x n) = sup min( A (x );:::; An (x n )) x ;:::;x n y=f(x ;:::;x n) (2.a) (2.b) where the Cartesian product A A n is used to represent the multi-dimensional fuzzy set A, because the fuzzy set A is usually not available. This implies A to be the

29 2 Fuzzy sets and relations largest set whose projections on X ;:::;X n are A ;:::;A n, respectively (Dubois et al., 993a). See section 2.4. for more details on projections. Hence, the extension principle allows the derivation of a fuzzy set B on y by B = f(a ;:::;A n ),wherea i are fuzzy sets in X i. In other words: the extension principle can be used to extend normal mathematical operations to operations with fuzzy sets. Another way to write (2.a) is: B = f(a ;:::;A n ) = Z X :::X n min( A (x );:::; An (x n ))=f(x ;:::;x n ) (2.c) (2.d) To clear things up, here is a small example. Suppose the addition of two fuzzy numbers is desired, then the extension principle can be used to derive a resulting fuzzy set for the outcome of the addition. Let us take the fuzzy number also used in section 2..3, about 5 with membership function 5 (x ) = max(, 2 jx, 5j; ), shown in figure 2.7a, and a fuzzy number to add to this, about 2, with membership function 2 (x 2 )= max(,jx 2, 2j; ), shown in figure 2.7b. about 2 about 5 x x 2 Figure 2.5: Cartesian product of about 5 and about 2. The Cartesian product of about 5, projected on an x-axis, and about 2, projected on a y-axis, is shown in figure 2.5. In figure 2.6, a contour plot of the Cartesian product space X X 2

30 2. Fuzzy sets 2 x2 y=8 y=9 y= y=4 y=5 y=6 y= x Figure 2.6: Contour plot for product space X X 2 with fuzzy sets 5 (x ) = max(, 2 jx, 5j; ) and 2 (x 2 ) = max(, jx 2, 2j; ). Isolines for y = 4; 5;:::; and isocurves for max( 5 (x ); 2 (x 2 )) = ; 2 ;:::; are shown.

31 22 Fuzzy sets and relations is shown with isolines for y. The rectangles represent isocurves of min( 5 (x ); 2 (x 2 )) for a number of membership values:, 4, 2, 3 4 and (which is only one point). The next step in applying the extension principle is to take the supremum of min( 5 (x ); 2 (x 2 )) for each isoline in order to obtain the membership value for B (y) with y = x + x 2.In figure 2.6 several of those lines are shown. The resulting membership function B (y) is max(, 3jy, 7j; ), shown in figure 2.7c. In figure 2.7, the calculus of the fuzzy numbers is shown schematically. When the universes are discrete, x i 2 Z, the example simplifies to the following: about 5 + about 2 =( 2 =4+=5 + 2 =6) + (=2) = 2 =6+=7 + 2 =8 + = x 23 x x 2 (a) (b) (c) 5 (x ) 2 (x 2 ) 5 (x )+ 2 (x 2 ) Figure 2.7: Addition of two fuzzy numbers using extension principle. Using triangular-shaped membership functions, the determination of the outcome is more or less trivial for addition and subtraction. However, in the case of more complexly shaped membership functions, the application of the extension principle can involve a rather severe calculational load due to calculus on product spaces. Also in the case of more complex functions or mathematical operations, the application of the extension principle often cannot be simplified. For example, calculation of the product instead of the addition of two fuzzy numbers will change the isolines for y in figure 2.6 into hyperbolic isocurves. The product of two fuzzy numbers with triangular-shaped membership functions will therefore not result in a triangular-shaped membership function for the outcome. From the extension principle it can be derived that an operation f() on fuzzy sets which are fuzzy subsets of the same universe can be written as (Kandel, 986): [ f(a ;A 2 )= f(-cut(a );-cut(a 2 )) (2.2) 2h;]